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Theorem madebdayim 27940
Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
Assertion
Ref Expression
madebdayim (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim
Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6943 . . 3 (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ dom M )
2 madef 27909 . . . 4 M :On⟶𝒫 No
32fdmi 6747 . . 3 dom M = On
41, 3eleqtrdi 2848 . 2 (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ On)
5 fveq2 6906 . . . . . 6 (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6 sseq2 4021 . . . . . 6 (𝑎 = 𝑏 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝑏))
75, 6raleqbidv 3343 . . . . 5 (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝑏)( bday 𝑥) ⊆ 𝑏))
8 fveq2 6906 . . . . . . 7 (𝑥 = 𝑦 → ( bday 𝑥) = ( bday 𝑦))
98sseq1d 4026 . . . . . 6 (𝑥 = 𝑦 → (( bday 𝑥) ⊆ 𝑏 ↔ ( bday 𝑦) ⊆ 𝑏))
109cbvralvw 3234 . . . . 5 (∀𝑥 ∈ ( M ‘𝑏)( bday 𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏)
117, 10bitrdi 287 . . . 4 (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏))
12 fveq2 6906 . . . . 5 (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
13 sseq2 4021 . . . . 5 (𝑎 = 𝐴 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝐴))
1412, 13raleqbidv 3343 . . . 4 (𝑎 = 𝐴 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴))
15 elmade2 27921 . . . . . . . 8 (𝑎 ∈ On → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
1615adantr 480 . . . . . . 7 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17 elpwi 4611 . . . . . . . . . . 11 (𝑙 ∈ 𝒫 ( O ‘𝑎) → 𝑙 ⊆ ( O ‘𝑎))
18 elpwi 4611 . . . . . . . . . . 11 (𝑟 ∈ 𝒫 ( O ‘𝑎) → 𝑟 ⊆ ( O ‘𝑎))
1917, 18anim12i 613 . . . . . . . . . 10 ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)))
20 unss 4199 . . . . . . . . . 10 ((𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)) ↔ (𝑙𝑟) ⊆ ( O ‘𝑎))
2119, 20sylib 218 . . . . . . . . 9 ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙𝑟) ⊆ ( O ‘𝑎))
22 simpr 484 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23 simplll 775 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24 dfss3 3983 . . . . . . . . . . . . . . . . 17 ((𝑙𝑟) ⊆ ( O ‘𝑎) ↔ ∀𝑧 ∈ (𝑙𝑟)𝑧 ∈ ( O ‘𝑎))
25 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 → ( bday 𝑦) = ( bday 𝑧))
2625sseq1d 4026 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → (( bday 𝑦) ⊆ 𝑏 ↔ ( bday 𝑧) ⊆ 𝑏))
2726rspccv 3618 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏))
2827ralimi 3080 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → ∀𝑏𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏))
29 rexim 3084 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑏𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏) → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3130adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
32 elold 27922 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏)))
3332adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏)))
34 bdayelon 27835 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝑧) ∈ On
35 onelssex 6433 . . . . . . . . . . . . . . . . . . . . 21 ((( bday 𝑧) ∈ On ∧ 𝑎 ∈ On) → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3634, 35mpan 690 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3736adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3831, 33, 373imtr4d 294 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) → ( bday 𝑧) ∈ 𝑎))
3938ralimdv 3166 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙𝑟)𝑧 ∈ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
4024, 39biimtrid 242 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙𝑟) ⊆ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
4140imp 406 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎)
4241adantr 480 . . . . . . . . . . . . . 14 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎)
43 bdayfun 27831 . . . . . . . . . . . . . . . . 17 Fun bday
44 oldssno 27914 . . . . . . . . . . . . . . . . . . 19 ( O ‘𝑎) ⊆ No
45 sstr 4003 . . . . . . . . . . . . . . . . . . 19 (((𝑙𝑟) ⊆ ( O ‘𝑎) ∧ ( O ‘𝑎) ⊆ No ) → (𝑙𝑟) ⊆ No )
4644, 45mpan2 691 . . . . . . . . . . . . . . . . . 18 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (𝑙𝑟) ⊆ No )
47 bdaydm 27833 . . . . . . . . . . . . . . . . . 18 dom bday = No
4846, 47sseqtrrdi 4046 . . . . . . . . . . . . . . . . 17 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (𝑙𝑟) ⊆ dom bday )
49 funimass4 6972 . . . . . . . . . . . . . . . . 17 ((Fun bday ∧ (𝑙𝑟) ⊆ dom bday ) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5043, 48, 49sylancr 587 . . . . . . . . . . . . . . . 16 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5150adantl 481 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5251adantr 480 . . . . . . . . . . . . . 14 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5342, 52mpbird 257 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday “ (𝑙𝑟)) ⊆ 𝑎)
54 scutbdaybnd 27874 . . . . . . . . . . . . 13 ((𝑙 <<s 𝑟𝑎 ∈ On ∧ ( bday “ (𝑙𝑟)) ⊆ 𝑎) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
5522, 23, 53, 54syl3anc 1370 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56 fveq2 6906 . . . . . . . . . . . . 13 ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘(𝑙 |s 𝑟)) = ( bday 𝑥))
5756sseq1d 4026 . . . . . . . . . . . 12 ((𝑙 |s 𝑟) = 𝑥 → (( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝑎))
5855, 57syl5ibcom 245 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → ( bday 𝑥) ⊆ 𝑎))
5958expimpd 453 . . . . . . . . . 10 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎))
6059ex 412 . . . . . . . . 9 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙𝑟) ⊆ ( O ‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎)))
6121, 60syl5 34 . . . . . . . 8 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎)))
6261rexlimdvv 3209 . . . . . . 7 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎))
6316, 62sylbid 240 . . . . . 6 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) → ( bday 𝑥) ⊆ 𝑎))
6463ralrimiv 3142 . . . . 5 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎)
6564ex 412 . . . 4 (𝑎 ∈ On → (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎))
6611, 14, 65tfis3 7878 . . 3 (𝐴 ∈ On → ∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴)
67 fveq2 6906 . . . . 5 (𝑥 = 𝑋 → ( bday 𝑥) = ( bday 𝑋))
6867sseq1d 4026 . . . 4 (𝑥 = 𝑋 → (( bday 𝑥) ⊆ 𝐴 ↔ ( bday 𝑋) ⊆ 𝐴))
6968rspccv 3618 . . 3 (∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴 → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
7066, 69syl 17 . 2 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
714, 70mpcom 38 1 (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cun 3960  wss 3962  𝒫 cpw 4604   class class class wbr 5147  dom cdm 5688  cima 5691  Oncon0 6385  Fun wfun 6556  cfv 6562  (class class class)co 7430   No csur 27698   bday cbday 27700   <<s csslt 27839   |s cscut 27841   M cmade 27895   O cold 27896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-no 27701  df-slt 27702  df-bday 27703  df-sslt 27840  df-scut 27842  df-made 27900  df-old 27901
This theorem is referenced by:  oldbdayim  27941  madebday  27952
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